If both the mass and the radius of the earth decrease by $1\%$,the value of the acceleration due to gravity will

If $x=a\left[\cos \theta+\log \left\{\tan \left(\frac{\theta}{2}\right)\right\}\right]$ and $y=a \sin \theta$,then $\frac{dy}{dx}$ is equal to

Two liquids $A$ and $B$ have vapour pressures in the ratio $P_A^0 : P_B^0 = 1 : 3$ at a certain temperature. If the ratio of mole fractions of $A$ to $B$ in the vapour phase is $4 : 3$,then the mole fraction of $B$ in the solution at the same temperature is

$A$ disc of radius $R$ is made to oscillate about a horizontal axis passing through its periphery. Its time period would be

The reaction sequence given below is carried out with $16$ moles of $X$. The yield of the major product in each step is given below the product in parentheses. The amount $($in grams$)$ of $S$ produced is $.....$
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Use: Atomic mass $($in amu$): H =1, C =12, O =16, Br =80$

The equation of the circle with radius $5$ and touching the coordinate axes in the third quadrant is